in which Gk=[Gk−1,Gk−1]G_k = [G_{k-1},G_{k-1}] is the commutator, that is the subgroup of Gk−1G_{k-1} generated by all elements of the form ghg−1h−1ghg^{-1}h^{-1} where g,h∈Gk−1g,h\in G_{k-1}. A group is solvable iff its derived series terminates with the trivial subgroup after finitely many terms.

Similarly, one defines a derived series for a Lie algebra LL, and for Ω\Omega-groups.